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A well-known puzzle involving the repeated flying of a swallow between two cyclists is extended to investigate the effect of nonuniform speeds. The resulting generalizations provide examples that can be used in the context of the teaching aspects of the manipulation of infinite series, including the use of telescoping sums and products. [ABSTRACT FROM AUTHOR]

The purpose of the study was to explore students' consistency in their notions encountering different methods in solving the same task of comparing values of certain infinite series. The participants were 93 male students aged 16–17. Applying cognitive conflict approach and the data were collected through a questionnaire administrated in two stages. In the first stage, students were exposed to a challenging question involving comparison of two infinite series. Then, they encountered all types of various responses consisting of different results provided by the participants, and they were asked to perform the same task again. Analysis of the data, focusing on the changes provided revealed two important points: (a) the students were generally consistent in their notions, even in their incorrect answers, and (b) the students' intuitive notions about infinite series were more complicated than those about infinite sets and sequences. [ABSTRACT FROM AUTHOR]